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For matrices written $H = L+W$, where $L$ is either a combinatorial or normalized graph Laplacian, I show that: (1) when $W$ is diagonal and $L$ has maximum degree $d_{\\max}$, $2h \\geq \\gamma \\geq \\sqrt{h^2 + d_{\\max}^2}-d_\\max$; (2) when $W$ is real, we can often route negative-weighted edges along positive-weighted edges such tha"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1804.06857","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2018-04-18T18:00:06Z","cross_cats_sorted":["math.CO","quant-ph"],"title_canon_sha256":"76331f6245c71e02119bda2cbf5327dbbbbfb5fb57594227a3cb10de40212b3d","abstract_canon_sha256":"b5ae65c7bec246e55195389652f4a6f4759a4f87da6af92a2a4e0f887ecbb0d5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:02:44.996542Z","signature_b64":"+r+9n4eoW3aSozizvVUHTkWO2QIAtdCNcr3TV/h3+WghqWRA8hOcOTfYsckbKNbIE5Hyr/7FcKyo+cVjpc1WCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8e9db2f1e9a5cd11cf2fe69b43ba8316f1ce21b49179d2163c11991d298f7ef5","last_reissued_at":"2026-05-18T00:02:44.995982Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:02:44.995982Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Hamiltonian surgery: Cheeger-type gap inequalities for nonpositive (stoquastic), real, and Hermitian matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","quant-ph"],"primary_cat":"math.SP","authors_text":"Michael Jarret","submitted_at":"2018-04-18T18:00:06Z","abstract_excerpt":"Cheeger inequalities bound the spectral gap $\\gamma$ of a space by isoperimetric properties of that space and vice versa. 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